1. Infinite Series
One important application of infinite sequences is in representing “infinite summations.”
Informally, if {an} is an infinite sequence, then
is an infinite series (or simply a series).
The numbers a1, a2, a3, are the terms of the series. For some series it is convenient to begin the index at n = (or some other integer). As a typesetting convention, it is common to represent an infinite series as simply

2. Infinite Series
In such cases, the starting value for the index must be taken from the context of the statement.
To find the sum of an infinite series, consider the following sequence of partial sums.
If this sequence of partial sums converges, the series is said to converge.

3. Infinite Series

4. Example 1(a) – Convergent and Divergent Series
The series
has many partial sums. Find

5. Example 1(a) – Convergent and Divergent Series
cont’d
Because
it follows that the series __________ and its sum is ___.

6. Example 1(b) – Convergent and Divergent Series
cont’d
The nth partial sum of the series
is given by
Because the limit of Sn is 1, the series _________ and its sum is ___.

7. Example 1(c) – Convergent and Divergent Series
cont’d
The series
_______ because Sn = n and the sequence of partial sums _______.

8. Infinite Series The series is a telescoping series of the form
Note that b2 is canceled by the second term, b3 is canceled by the third term, and so on.

9. Infinite Series Because the nth partial sum of this series is
Sn = b1 – bn + 1
it follows that a telescoping series will converge if and only if bn approaches a finite number as
Moreover, if the series converges, its sum is

10. Geometric Series The series is a geometric series.
In general, the series given by
is a geometric series with ratio r.

11. Geometric Series

12. Because 0 < |r| < 1, the series __________ and its sum is
Example 3(a) – Convergent and Divergent Geometric Series
The geometric series
has a ratio of with a = .
Because 0 < |r| < 1, the series __________ and its sum is

13. Because ______, the series _________.
Example 3(b) – Convergent and Divergent Geometric Series
cont’d
The geometric series
has a ratio of
Because ______, the series _________.

14. Geometric Series

15. nth-Term Test for Divergence
The contrapositive of Theorem 9.8 provides a useful test for divergence. This nth-Term Test for Divergence states that if the limit of the nth term of a series does not converge to 0, the series must diverge.

16. Example 5 – Using the nth-Term Test for Divergence
a. For the series you have
So, the limit of the nth term is _____, and the series
_______.
b. For the series you have
So, the limit of the nth term is ______, and the series
_________.

17. Example 5 – Using the nth-Term Test for Divergence
cont’d
c. For the series you have
Because the limit of the nth term is ___, the nth-Term Test for Divergence does not apply and you can draw no conclusions about convergence or divergence.
In other words, the Divergence Test can only help you determine Divergence!! To determine convergence, find another test. In the next five sections of chapter 9, you will be learning many tests that help you determine convergence like the Geometric Series Test.